13.7 Tangent Planes and Normal Lines
for an animation of this topic visit
http://www.math.umn.edu/~rogness/multivar/tanplane_withvectors.shtml
Recall from chapter 11:
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Standard equation of a plane in Space
a(x-x1) + b(y-y1) + c (z – z1) = 0
parametric form equations of a line in
space:
x = x1 + at
y = y1 +bt
z = z1 +ct
symmetric form of the equations of a line
in space
x-x1 = y – y1 = z – z1
a
b
c
Example 1
For the function f(x,y,z) describe the level surfaces
when f(x,y,z) = 0,4 and 10
Example 1 solution
For the function f(x,y,z) describe the level surface
when f(x,y,z) = 0,4 and 10
For animated normal vectors visit:
http://www.math.umn.edu/~rogness/math2374/paraboloid_normals.html
OR
http://www.math.umn.edu/~rogness/multivar/conenormal.html
Example 2
Find an equation of the tangent plane to given the
hyperboloid at the point (1,-1,4)
Example 2 Solution:
Example 3
Find the equation of the tangent to the given
paraboloid at the point (1,1,1/2)
Example 3 Solution: Find the equation of the
tangent to the given paraboloid at the point
(1,1,1/2). Rewrite the function as
f(x,y,z) =
-z
Example 4
Find a set of symmetric equations for the
normal line to the surface given by
xyz = 12
At the point (2,-2,-3)
Example 4 Solution
Find a set of symmetric equations for the normal
line to the surface given by
xyz = 12 At the point (2,-2,-3)
One day in my math class, one of my students spent the entire period standing
leaning at about a 30 degree angle from standing up straight.
I asked her “Why are you not standing up straight? “
She replied “Sorry, I am not feeling normal.”
Of course that students name was Eileen.
- Mr. Whitehead